Tunneling splittings in model 2D potentials. II. V(X, V) = λ(X2 − X02)2 − CX2(Y − Y0) + ½Ω2(Y − Y0 + CX02/Ω2)2 − C2X04/2Ω2 Original Research Article

We propose a method for calculating semiclassical wave functions of the vibrationally excited states in model 2D potentials with two degenerate minima. The procedure includes two steps. First, the approximate solution of the Schrödinger equation is found variationally in the vicinity of one of the minima. This solution is made separable along effective vibrations, other than the actual anharmonic ones. The parameters of the effective vibrations depend on quantum numbers of a given stationary state. Second, the Hamilton—Jacobi and transport equations are solved in the classically forbidden region in the approximation of small fluctuations around the extreme tunneling path, and the “tunneling” WKB wave function is constructed. Due to the freedom in decomposition of the total energy between the WKB equations, this wave function is extended to the potential minimum, where its form coincides with the variational one. This enables one to correctly normalize the WKB wave function avoiding matching at the caustics. Application of the proposed method to calculating tunneling splittings of the vibrationally excited states in the “gated” and “squeezed” potentials with different sets of parameters (corresponding, in particular, to hydrogen transfer in malonaldehyde molecule and puckering of cyclopentanone molecule) gives results that agree well with the quantum mechanical results obtained by diagonalization of Hamiltonian matrices.